MI 3
March 9, 2012
2 pts.
Poly Unit Test
Name: ______________
Show all work! Calculators are not allowed.
State whether the following statements are true or false. Briefly explain your reasoning.
False 1) y 2( x 5)2 3 has two real zeros.
y 0 for all x since ( x 5)2 0 for all x . Therefore this quadratic has no real roots.
2 pts.
False 2) If a polynomial with real coefficients has a root of x 2 i , then ( x (2 i )) is a factor
of the polynomial.
The factors would be ( x (2 i )) and ( x (2 i ))
2 pts.
False 3) The graph of the polynomial y x5 3x3 6 x 5 could be as shown below:
This is the graph of an even degree polynomial since y , for x and x
4) Write the equation of a seventh degree polynomial, with integer coefficients, with a bounce
point at x 3 , a pass- through point at x 6 , and an imaginary root at 1 i . Write out your
polynomial equation in factored form.
One possibility is: f ( x) ( x 3)4 ( x 6)( x (1 i))( x (1 i))
MI-3
v1
SP ‘12
2 each
5) Determine whether each of the following is a polynomial. If it is a polynomial,
state its degree, if not a polynomial, explain why.
3
g ( x) 6 x 2 2 2 Not a polynomial since the last term has x raised to a negative
x
2
power, 3x .
k ( x) 3 x8 5 x 7 1
Is a polynomial of degree 8.
3 pts.
6) Evaluate (5 3i) (2 i) .
(5 3i) (2 i) (3 4i) 32 42 5
4 pts.
7) Solve for z and express the answer in a bi form:
(3 2i ) z (4 i) 2 3i (3 2i) z 2 4i
2 4i
z
(3 2i )
(2 4i ) (3 2i)
z
(3 2i ) (3 2i )
(14 8i)
z
13
2 pts.
8) Find the sum and the product of the roots for the given polynomial without actually finding
its roots:
g ( x) 2 x5 8 x 4 5 x 2 14
Sum =
8
4
2
Product =
14
7
2
9) Write the equation of a polynomial function of smallest degree with the graph given below.
y a( x 1)3 ( x 2)2 ( x 3) . Since
y(0) 48 a 1 4 (3), so a 4 y 4( x 1)3 ( x 2)2 ( x 3)
48
1
10) Sketch the graph of p ( x)
3 pts.
2
3
1
( x 1) 2 ( x 2)3 ( x 4) . Indicate x and y intercepts clearly.
2
11) Given f ( x) ( x 1)( x 2)( x 1) . Find c so that 2 is a zero of y f ( x) c .
If 2 is a root of y f ( x) c , then 0 f (2) c 0 (2 1)(2 2)(2 1) c c 12
4 pts.
f x 2x3 3x2 4x 15
12) Determine the exact values, over the
complex numbers, of all the zeros of :
y
f x 2x3 3x2 4x 15
Show your work clearly.
x
4 pts.
13) Solve z 3 27 0 and graph and label the solutions clearly.
z
im aginary
real
0
